To determine how long it takes for an initial amount of [tex]$50 to grow to $[/tex]250 at an annual interest rate of 4% compounded continuously, we can use the continuous compounding formula:
[tex]\[ A = P \cdot e^{rt} \][/tex]
Where:
- [tex]\( A \)[/tex] is the final amount ([tex]$250)
- \( P \) is the initial amount ($[/tex]50)
- [tex]\( r \)[/tex] is the annual interest rate (4% or 0.04 as a decimal)
- [tex]\( t \)[/tex] is the time in years
- [tex]\( e \)[/tex] is the base of the natural logarithm (approximately equal to 2.71828)
We need to solve for [tex]\( t \)[/tex] in the equation:
[tex]\[ 250 = 50 \cdot e^{0.04t} \][/tex]
First, isolate the exponential term [tex]\( e^{0.04t} \)[/tex]:
[tex]\[ \frac{250}{50} = e^{0.04t} \][/tex]
[tex]\[ 5 = e^{0.04t} \][/tex]
Next, take the natural logarithm (ln) of both sides to remove the exponential:
[tex]\[ \ln(5) = \ln(e^{0.04t}) \][/tex]
Since [tex]\( \ln(e^x) = x \)[/tex], this simplifies to:
[tex]\[ \ln(5) = 0.04t \][/tex]
Now solve for [tex]\( t \)[/tex] by dividing both sides by 0.04:
[tex]\[ t = \frac{\ln(5)}{0.04} \][/tex]
Using the natural logarithm of 5:
[tex]\[ \ln(5) \approx 1.6094 \][/tex]
Thus:
[tex]\[ t = \frac{1.6094}{0.04} \][/tex]
[tex]\[ t \approx 40.235 \][/tex]
Rounding to the nearest whole number:
[tex]\[ t \approx 40 \][/tex]
So, it will take approximately 40 years for [tex]$50 to grow to $[/tex]250 at a 4% annual percentage rate when compounded continuously.
